The 30-year-old coach was added to the backroom staff by Ole Gunnar Solskjaer last year and he served under interim boss Ralf Rangnick as well.
Manchester United want to keep Ramsay and want him to be part of Erik ten Hag’s staff at Old Trafford going forward.
However, he is one of the names Barnsley are considering as they look to bring in a new manager ahead of next season.
And it has been claimed that Manchester United will hold talks with Ramsey if the Oakwell outfit make an approach.
Ten Hag wants to keep him and Manchester United are hoping to convince Ramsay to stick around at Old Trafford.
However, the Red Devils did not stand in Kieran McKenna’s way when he was approached by Ipswich Town in December last year.
Ramsay is likely to be tempted by the opportunity of starting his career at the League One club next season.
Ten Hag has brought in Mitchell van der Gaag and Steve McClaren as his assistants at Manchester United.
